1. Field of the Invention
The present invention relates to an ohmmeter of the type capable of measuring small electric resistance in units of milliohms.
2. Description of the Prior Art
Electrical resistance is the rate of the potential difference between the ends of a conductor to be measured to the electrical current flowing in the conductor. With respect to FIG. 1, which is a diagram of a model circuit showing the relationship of electrical current I with the voltage difference V across a resistor R.sub.x is expressed by the formula: EQU R=V/I (1)
In the diagram, e is an alternating current source generating a current of known value I, V.sub.o is a voltmeter giving readings of voltage difference V, and R.sub.x is a resistor or conductor supposed to have a resistance R to be measured.
With the progress of precise electrical applications, it has become increasingly necessary to determine very small magnitudes of resistance.
Measurements of resistance in minute units naturally involves dealing with correspondingly smaller denominations of amperage and voltage. Measurements of infinitesimal magnitudes have normally involved using attenuators and amplifiers with high and stable gains, such as operational amplifiers.
However, these conventional methods have been found to pose problems. The operational amplifiers have generated noises having serious effects on the readings.
Some attempts have been made to eliminate the above drawback. For example, as in most cases, a noise filter has been connected to the amplifier. However, the use of noise filters have failed to help much.
Assuming that an ideal measuring instrument is used to determine the voltage difference V across a conductor with a resistance of I microhm when an alternating current at 1 milliampere is applied to the conductor, the theoretical voltage difference V should be 1 nanovolt, or 1.times.10.sup.-9 =1.times.10.sup.-6 .times.1.times.10.sup.-3, according to the equation V=RI obtained by adjusting the equation 1 above for V. However, in actual practice, the generation of noises must be considered. Using statistical mechanics, which looks into the mechanical properties of large assemblies of particles or components in terms of statistics, noise Vn in volts for voltage difference nV to be measured is given by the formula: EQU Vn=2.sqroot.kTR.DELTA.fnV (2)
where K is Boltzmann's constant, T is absolute temperature, R is conductor resistance and .DELTA.f is the noise range of the instrument used for measuring voltage differences.
Even when one of those most advanced operational amplifier producing the least noises is used, for a very small voltage difference of 1.times.10.sup.-9, the amplifier would develop a noise level of about 1.2.sqroot..DELTA.fnV, a level almost approaching the difference to be measured, using the equation 2 above, where 2/kTR is known to be 1.2.times.10.sup.-9. If the noises generated are that high, accurate measurements of resistances through amplification of voltage differences would almost be impossible, particularly when magnitudes involved are very small.
Various improvements have been proposed to eliminate the above difficulty. For example, a phase detection circuit 8, as illustrated in schematic form in FIG. 2, may be used. The phase detection circuit 8 comprises an inverter circuit 10 and a R-C circuit 12.
With respect to FIG. 2, when a voltage difference Vo to be measured is applied to the detection circuit 8 at its input terminal 8a, the inverter circuit 10 splits each cycle of the input signal Vo into a first and a second half wave. The first halves of positive polarity of the signal Vo appear, as they are, on a first output terminal 8b of the inverter circuit 10. A combination of a operational Amplifier A and a pair of resistors R.sub.1 and R.sub.2 inverts the second halves in polarity, and outputs the inverted second half waves through a second output terminal 10c. A switch means S alternates between the first output terminal 10b and second output terminal 10c to integrate the noniverted first halves and inverted second halves into a train of whole wave cycles. In this process, noises are superimposed onto the voltage signal Vo being measured. The R-C circuit 12, which is comprised of a capacitor C.sub.12 and a resistor R.sub.12, is connected to the inverter circuit 10 to achieve a reduction in the noise signal superimposed over the voltage signal to be measured. The R-C circuit 12 outputs the result of electric charge storage at its output terminal 12h as a voltage level e.sub.o, after noise reduction, which would be a close approximation of the voltage signal Vo to be measured. The process of this reduction will be explained in more detail with reference to the diagrams in FIG. 3.
The diagrams shown in FIG. 3 are the waveform of difference voltage signals, with times taken along the horizontal axis. FIG. 3(a) is the waveform of the voltage difference Essin .omega.t to be measured. The waveform of FIG. 1 is superimposed onto the waveform of the noise Ensin (.alpha.t+.theta.) generated, as shown in FIG. 3(b), to develop a combined waveform Essin .omega.t Ensin (.alpha.t+.theta.), as indicated in FIG. 3(c). In this description, the noise is shown as a sine wave signal for the sake of simplicity of explanation, although noises in actual cases come in far more complicated waveforms. On the output terminal of the inverter circuit 10, a superimposed signal similar to the one in FIG. 3(d) would appear as a result of inversing the second half of each wave cycle in the waveform of FIG. 3(c).
The inversed voltage signal of FIG. 3(d) is expressed as E.sub.s sin .omega.t+E.sub.n sin (.alpha.t+.theta.) when (2n-1).pi./.omega.&gt;t2(n-1)/.omega., and -{E.sub.s sin .omega.t+E.sub.n sin (.alpha.t+.theta.)} when 2n.pi./.omega.&gt;t&gt;2(n-1).pi./.omega., where n: integer, .omega.: frequency, .alpha.: frequency of noise voltage, .theta.: phase of noise voltage.
Since the signal shown in FIG. 3(d) is inversed at time intervals .pi./.omega., that part of it which constitutes the voltage signal Vo to be measured is expressed as Essin .omega.t in absolute value.
However, although the part of the inversed FIG. 3(d) signal that constututes the noise voltage, which is also inversed at time intervals of .pi./.omega., its time average would come almost zero since such noise can normally occur in wide ranges of amplitude and frequency.
In normal condition, the time average of the output voltage signal e.sub.o from the inverter circuit 10 of FIG. 2 is expressed as 2/.pi..multidot.Es.
It might be important to note here that, instead of inverting the voltage signal Vo to be measured in the manner as described above, full-wave rectification of the superimposed voltage of FIG. 3(c) to obtain a waveform as shown in FIG. 3(e) would mean irrelevance. Such full-wave rectification would produce an average of Essin .omega.t+Ensin (.alpha.t+.theta.) in absolute value, including the noise signal component Essin (.alpha.t+.theta.)
The phase detection circuit 8 shown in FIG. 2 yields an average of the inversed voltage signal of FIG. 3(d) through the function of the capacitor C.sub.12.
The voltage signal that appears on the output terminal of the capacitor C.sub.12 comes Essin .omega.t, an absolute value, since the noise component is reduced to zero through integration by the capacitor.
Letting the voltage signal e.sub.i input to the phase detection circuit 8 Essin .omega.t, the voltage signal e.sub.o appearing on the output terminal of the capacitor C.sub.12 is given by a series of computations as follows:
When the frequency .omega. for the signal is .pi./.omega.&gt;t&gt;0, e.sub.o is expressed as follows using the Laplace transformation: EQU e.sub.o =Es/(1+.omega..sup.2 C.sup.2 R.sup.2){(sin .omega.t-.omega.CR cos .omega.t)+.omega.CRe.sup.-t/RC } (3)
when
C: capacitance value of capacitor, C.sub.12 PA1 R: resistance value of resistor R.sub.12 PA1 e: euler's constant PA1 fo: the frequency of the voltage signal being measured, which is equivalent to .omega./2.pi. for the aforesaid voltage signal Essin .omega.t PA1 .tau.=CR: time constants by C.sub.12, R.sub.12 in the circuit of FIG. 2 PA1 E.sub.n : the amplitude of the noise signal
When 2.pi./.omega..gtoreq.t.gtoreq..pi./.omega., e.sub.o is expressed using the theorem of superimposition: EQU e.sub.o =Es/(1+.omega..sup.2 C.sup.2 R.sup.2){-(sin .omega.t-.omega.CR cos .omega.t)+.omega.CR(1+2e.sup..pi./.omega.CR)e.sup.-t/RC } (4)
Generally, in the relation (n-1).pi./.omega.&lt;t&lt;n.pi. to determine the value for n, fthe voltage signal e.sub.o is:
(1) for the value "n" being an odd number, EQU e.sub.o =Es/(1+.omega..sup.2 C.sup.2 R.sup.2){(sin .omega.t-.omega.CR cos .omega.t)+.omega.CR(1+2e.sup..pi./.omega.CR +2e.sup.2.pi./.omega.CR . . . +2e.sup.(n-1).pi./.omega.CR)e.sup.-t/RC },
and, (2) for the value "n" being an even number, EQU e.sub.o =Es/(1+.omega..sup.2 C.sup.2 R.sup.2){-(sin .omega.t-.omega.CR cos .omega.t)+.omega.CR(1+2e.sup..pi./.omega.CR +2e.sup.2.pi./.omega.CR . . . +2e.sup.(n-1) .pi./.omega.CR)e.sup.-t/RC },
Combining the equations (5) and (6) above gives: EQU e.sub.o =Es/(1+.omega..sup.2 C.sup.2 R.sup.2)[.+-.(sin .omega.t-.omega.CR cos .omega.t)+.omega.CR{(2e.sup.n.pi./.omega.CR -2)/(e.sup..pi./.omega.CR -1)-1}e.sup.-t/RC ] (7)
Of the double signs .+-. in the equation 7 above, the "+" is for the value "n" being an odd numer, and the "-" is for the value "n" being an even number.
In actuality, since voltage signals to be measured normally are designed greater in frequency .omega. and time constants R, C, so .omega.RC&gt;1.
Thus, the last term in the righthand side of the equation 7 above can be simplified as follows: ##EQU1##
For the equation 8 above, e.sup.n.pi./.omega.CR-t/RC .apprxeq.1, e.sup..pi./.omega.CR-t/RC .apprxeq.1. EQU e.sup..pi./.omega.CR .apprxeq.1+.pi./.omega.CR
Putting the value for the above-mentioned term into the equation 7 for the capacitor output voltage signal e.sub.o, and further approximation yields: EQU e.sub.o =2Es/.pi.(1-e.sup.-t/RC) (9)
In the above computation, the value for the first term of the equation 7 is ignored because it is very small compared with the value for the last term.
In other words, the value of the transient voltage signal e.sub.o appearing on the output terminal of the capacitor C.sub.12 of FIG. 2 is obtained by multiplying the term "1-e.sup.-t/RC " of the equation 7 with the average value 2Es/.pi. that is reached by full-wave rectification of the voltage signal input to the phase detection circuit 8.
Accordingly, when the inversed voltage signal ein having value .vertline.Essin .omega.t.vertline. appearing on the output terminal of the phase detection circuit 8 in FIG. 2 is applied to the capacitor C.sub.12, the system would always take a delay of fixed duration before the output terminal of the capacitor produces constant voltage signals whose values are or close 2Es/.pi.. In this case, the noise voltage is ignored since a time average of its value becomes almost zero.
Furthermore, it is well known that the noise to signal ratio (S/N), the ratio of the magnitude of the signal to that of the noise, for the circuit in FIG. 2 is expressed by the following formula: EQU S/N=2/.tau.foEs/EN, (10)
where
For example, when the voltage signal is input to the phase detection circuit at a frequency of 1 kHz, the amplitude of the noise voltage En that would be produced even by one those most advanced amplifiers with the least noise is as follows using the equation 2 above for noise voltages: EQU E.sub.n =1.2.times./1,000.apprxeq.38 nanovolts.
Normally, the value for fo is 1 kHz. When the value from the equation 9 for S/N ratios is required to be 1, when the above value for fo, along with the aforesaid value for En is put into the equation, the value for .tau. must be 0.36 (in seconds). This means that the R-C circuit 12 must be designed such that the combined time constant for the capacitor C.sub.12 and resistor R.sub.12 is 0.36.
It thus follows that for an R-C circuit such as shown in FIG. 2 to produce its output voltage within a 99% target accuracy range of the voltage signal Es to be measure, the value for e.sup.-t/.tau. must be 0.01, based on the relation Es(1-e.sup.-t/.tau.) =0.99E.
This means that the operator has to wait about 1.7 seconds before the result of measurement at about 1% error is obtained.